It does sound weird. The tag question construction when formed from a clause with a [be] auxiliary or copula needs a subject pronoun to stand in. Not surprisingly, all the top hits in COCA are for the subject pronouns it, you, they, he, we, she, they. We also see expletive there showing up frequently when the tag is formed off of an existential there construction. There are, however, two hits that merit some scrutiny, isn't that and isn't this.

, IS N'T IT ? 4627
, ARE N'T YOU ? 1214
, WAS N'T IT ? 766
, ARE N'T THEY ? 512
, IS N'T HE ? 413
, ARE N'T WE ? 252
, IS N'T SHE ? 244
, WERE N'T YOU ? 196
, WAS N'T HE ? 191
, IS N'T THERE ? 156
, ARE N'T I ? 147
, WERE N'T THEY ? 114
, WAS N'T SHE ? 89
, ARE N'T THERE ? 49
, WAS N'T I ? 38
, WAS N'T THERE ? 37
, WERE N'T WE ? 29
, WERE N'T THERE ? 10
, ARE N'T YA ? 9
, IS N'T THAT ? 1
, WERE N'T I ? 2
, IS N'T THIS ? 1

Both of these come from talkshow transcripts, where people are speaking fast and constantly interrupting each other.

When we weed out the spurious uses of isn't that, one remains:

That's OK. And we are talking now about a very modest goal of trying to get over 70 Democratic votes in the House out of 211. That's what we're talking about , is n't that ? You think you'll make over 70?

And the instance of isn't this is not spurious either:

MR-GREGORY: Well, let me, let me, Secretary Duncan, let's, let's -- what do you see... SEC'Y-DUNCAN: Right. MR-GREGORY:... in the sense of the overall here? Because this is really the core , is n't this ? SEC'Y-DUNCAN: That's right. It is the core.

I wouldn't be surprised to find in very casual or incautious speech something like

My daughter struggles with math, but yours isn't having any problems, isn't yours?

I can imagine a sentence like: "My dog is a dachshund, isn't yours?" but this would probably not technically qualify as a tag question.
We would absolutely never say, "My dog is beautiful, isn't mine?"